To solve this problem, let's start with the given points $A(1, 2)$, $B(5, 1)$, and $C(3, -2)$ on the Cartesian plane.

3.3.1: Determine the coordinates of point $D$ to create parallelogram $ABCD$

In a parallelogram, the diagonals bisect each other. Thus, the midpoint of diagonal $AC$ should equal the midpoint of diagonal $BD$.

1. Calculate the midpoint of $AC$: $\text{Midpoint of } AC = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{1 + 3}{2}, \frac{2 - 2}{2} \right) = \left( 2, 0 \right)$

2. Let the coordinates of $D$ be $(x_D, y_D)$. We now calculate the midpoint of $BD$: $\text{Midpoint of } BD = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) = \left( \frac{5 + x_D}{2}, \frac{1 + y_D}{2} \right)$

3. Set the midpoints equal: $\left( \frac{5 + x_D}{2}, \frac{1 + y_D}{2} \right) = (2, 0)$

   This gives us two equations: $\frac{5 + x_D}{2} = 2 \quad \text{(1)}$ $\frac{1 + y_D}{2} = 0 \quad \text{(2)}$

4. Solve for $x_D$ and $y_D$: From equation (1): $5 + x_D = 4 \implies x_D = 4 - 5 = -1$

   From equation (2): $1 + y_D = 0 \implies y_D = -1$

Thus, the coordinates of point $D$ are $D(-1, -1)$.

3.3.2: Prove that $ABCD$ is not a rhombus

To prove that $ABCD$ is not a rhombus, we need to show that not all sides are of equal length.

1. Calculate the lengths of the sides:

   • Length of $AB$: $AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(5 - 1)^2 + (1 - 2)^2} = \sqrt{16 + 1} = \sqrt{17}$

   • Length of $BC$: $BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(3 - 5)^2 + (-2 - 1)^2} = \sqrt{4 + 9} = \sqrt{13}$

   • Length of $CD$: $CD = \sqrt{(x_D - x_C)^2 + (y_D - y_C)^2} = \sqrt{(-1 - 3)^2 + (-1 + 2)^2} = \sqrt{16 + 1} = \sqrt{17}$

   • Length of $DA$: $DA = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2} = \sqrt{(1 + 1)^2 + (2 + 1)^2} = \sqrt{4 + 9} = \sqrt{13}$

2. Compare the lengths:

   • $AB = \sqrt{17}$
   • $BC = \sqrt{13}$
   • $CD = \sqrt{17}$
   • $DA = \sqrt{13}$

Since $AB \neq BC$, the lengths are not all equal, proving that $ABCD$ is not a rhombus.

Summary:

• Coordinates of point $D$: $D(-1, -1)$
• Proof that $ABCD$ is not a rhombus: $AB$ and $CD$ are equal, but $BC$ and $DA$ are equal and not equal to $AB$.

If you need further details or have questions, let me know! Here are some related questions you might consider:

1. What are the coordinates of the midpoint of diagonal $BD$?
2. How do you calculate the area of parallelogram $ABCD$?
3. What conditions must be met for a quadrilateral to be a rhombus?
4. Can you derive the slope of each side of the parallelogram?
5. What other types of quadrilaterals can be formed with points $A$, $B$, $C$, and $D$?

Tip: Always check the lengths of the sides to determine the properties of quadrilaterals.